Yes they should.
If they are separated without a proper divorce they should not remarry though, they are still married. That would be adultery. This is the difference between being put away (which God hates and Jesus preached about) and having a writ of divorcement.
I can elaborate if you like, though I don't have time to now.
Have you been reading the 'divorcehope' website?
Their line of reasoning doesn't make sense in context. In Matthew 19, Jesus is talking about putting away one's wife WITH a certificate of divorce. They would use 'put away' to refer to LEGAL divorces, too. This is obvious from the context.
Jesus said in the passage that whoever puts away his wife, except it be for fornication, and marries another commits adultery. And He that marries her that is divorced commits adultery.
Let's consider the idea that 'put away' refers to putting away without a certificate. That doesn't make sense at all considering the previous verses. But let's suppose it's true. Then what is the point of the saying? The conclusion one would have to draw is that if your wife commits fornication, it's okay to up and marry another woman without giving a certificate to the first one.
And then the disciples get bent out of shape in the next verse? What? If you have to shell out the money to buy a little piece of parchment to divorce your wife, then it's better not to get married at all.
That's a pretty ridiculous scenario. Why would the disciples say it is better not to marry if the point was that you had to give a wife a certificate if you were going to put her away? That makes no sense. The law already taught about giving a writing of divorcement. The Pharisees taught that.
This interpretation seems to be based on made-up cultural history, with no historical evidence. The site that promotes it aren't scholarly in how they argue their case if they do have any evidence. If you dig around in the Mishneh/Talmud writings you can see that there was a debate over whether a man could put away his wife for any cause. One of the scholars (which they call 'rabbi' though only Christ is Rabbi) said that a man could put away his wife for burning the food. One allowed for putting away one's wife if one wants a younger one. The house of Shammai allowed for divorce for serious offenses like adultery. The house of Hillel was another influential house that was more liberal as long as there was a certificate.
So the issue mentioned in the passage is whether a man may put away his wife 'for any cause.' What did Jesus think about the 'any cause' divorce? Can you put away your wife if she burns the bread or does something you don't like? Jesus' said "except it be for fornication" which lines up with the line of reasoning you can read about in the Talmud. Mark's gospel leaves out the fornication reason.
A conservative viewpoint is that 'except it be for fornication' refers to sexual immorality before marriage. A broader conservative viewpoint includes a wide range of sexual immorality.
Greek speakers in the first few hundred years of Christianity took a very conservative view of the passage. I'm suspicious that people who post unscholarly websites know enough Greek to debunk all the Greek speakers and scholars throughout the ages, that it's a big conspiracy on the part of translators, and that Jesus was really loose and liberal on divorce. It's more likely that people are trying to make the Bible conform with contemporary (im)morality.